Molecular Symmetry -- Contents -- Preface -- 1 Symmetry Elements and Operations -- 1.1 Introduction -- 1.2 Symmetry Elements and Operations -- 1.2.1 Proper Rotations: Cn -- 1.2.2 The Plane of Symmetry: σ -- 1.2.3 The Inversion Centre: i -- 1.3 Examples of the Impact of Geometric Symmetry on Chemistry -- 1.3.1 Oxygen Transfer via Metal Porphyrins -- 1.3.2 Nuclear Magnetic Resonance: Chemical Equivalence -- 1.4 Summary -- 1.5 Self-Test Questions -- Further Reading -- 2 More Symmetry Operations and Products of Operations -- 2.1 Introduction -- 2.2 Background to Point Groups -- 2.3 Closed Groups and New Operations -- 2.3.1 Products of Operations -- 2.3.2 Fixed Symmetry Elements -- 2.3.3 The Final Missing Operation, Improper Rotations: Sn -- 2.3.4 Equivalences for Improper Rotation Operations -- 2.4 Properties of Symmetry Operations -- 2.4.1 Equivalent Operations and Equivalent Atoms -- 2.4.2 The Inverse of an Operation -- 2.4.3 The Order of the Product -- Operations that Commute -- 2.5 Chirality and Symmetry -- 2.6 Summary -- 2.7 Completed Multiplication Tables -- 2.8 Self-Test Questions -- 3 The Point Groups Used with Molecules -- 3.1 Introduction -- 3.2 Molecular Classification Using Symmetry Operations -- 3.3 Constructing Reference Models with Idealized Symmetry -- 3.4 The Nonaxial Groups: Cs,Ci,C1 -- 3.4.1 Examples of Molecules for the Nonaxial Groups: Cs,Ci,C1 -- 3.5 The Cyclic Groups: Cn, Sn -- 3.5.1 Examples of Molecules for the Cyclic Groups: Cn, Sn -- 3.6 Axial Groups Containing Mirror Planes: Cnh and Cnv -- 3.6.1 Examples of Molecules for Axial Groups Containing Mirror Planes: Cnh and Cnv -- 3.7 Axial Groups with Multiple Rotation Axes: Dn, Dnd and Dnh -- 3.7.1 Examples of Axial Groups with Multiple Rotation Axes: Dn,Dnd and Dnh -- 3.8 Special Groups for Linear Molecules:C v and Dh -- 3.9 The Cubic Groups: Td and Oh.

3.10 Assigning Point Groups to Molecules -- 3.11 Example Point Group Assignments -- 3.11.1 Example 1: Conformations of Cyclohexane -- 3.11.2 Example 2: Six-Coordinate Metal Complexes -- 3.12 Self-Test Questions -- 4 Point Group Representations, Matrices and Basis Sets -- 4.1 Introduction -- 4.2 Symmetry Representations and Characters -- 4.2.1 Water, H2O, C2v -- 4.2.2 Direct Products -- 4.3 Multiplication Tables for Character Representations -- 4.4 Matrices and Symmetry Operations -- 4.5 Diagonal and Off-Diagonal Matrix Elements -- 4.5.1 Ammonia, NH3, C3v -- 4.6 The Trace of a Matrix as the Character for an Operation -- 4.7 Noninteger Characters -- 4.7.1 Boron Trifluoride, BF3, D3h -- 4.8 Reducible Representations -- 4.8.1 Water, H2O, C2v -- 4.9 Classes of Operations -- 4.9.1 [Ni(CN)4]2−, D4h -- 4.10 Degenerate Irreducible Representations -- 4.10.1 Ammonia, NH3, C3v -- 4.11 The Labelling of Irreducible Representations -- 4.12 Summary -- 4.13 Completed Tables -- 4.14 Self-Test Questions -- Further Reading -- 5 Reducible and Irreducible Representations -- 5.1 Introduction -- 5.2 Irreducible Representations and Molecular Vibrations -- 5.3 Finding Reducible Representations -- 5.4 Properties of Point Groups and Irreducible Representations -- 5.5 The Reduction Formula -- 5.5.1 Applying the Reduction Formula -- 5.6 A Complete Set of Vibrational Modes for H2O -- 5.7 Choosing the Basis Set -- 5.7.1 Carbonyl Stretching Modes of [Fe(CO)5], D3h -- 5.8 The d-Orbitals in Common Transition Metal Complex Geometries -- 5.8.1 Square Planar, D4h -- 5.8.2 Tetrahedral, Td -- 5.8.3 Octahedral, Oh -- 5.8.4 Trigonal Bipyramidal, D3h -- 5.9 Linear Molecules: Groups of Infinite Order -- 5.10 Summary -- 5.11 Self-Test Questions -- 6 Applications in Vibrational Spectroscopy -- 6.1 Introduction -- 6.2 Selection Rules -- 6.2.1 Infrared Spectroscopy.

6.2.2 Infrared Absorption and the Greenhouse Gases -- 6.2.3 Interstellar H2 -- 6.2.4 Raman Spectroscopy -- 6.2.5 Comparison of Infrared and Raman Selection Rules -- 6.3 General Approach to Analysing Vibrational Spectroscopy -- 6.3.1 Example: the CH Stretch Bands of 1,4-Difluorobenzene -- 6.4 Symmetry-Adapted Linear Combinations -- 6.5 Normalization -- 6.6 The Projection Operator Method -- 6.6.1 Projection Operator Applied to the CH Stretches of 1,4-Difluorobenzene -- 6.6.2 The Projection Operator and Degenerate Representations -- 6.7 Linking Results for Symmetry-Inequivalent Sets of Atoms -- 6.7.1 Sets of Atoms Differing in Mass or Chemical Bond Strength -- 6.8 Additional Examples -- 6.8.1 Benzene, D6h -- 6.8.2 The fac and mer Isomers of Transition Metal Complexes -- 6.9 Summary -- 6.10 Self-Test Questions -- Further Reading -- 7 Symmetry in Chemical Bonding -- 7.1 Introduction -- 7.1.1 Wave Phenomena and Interference -- 7.1.2 The Born Interpretation of the Wavefunction -- 7.2 Bond Energies -- 7.2.1 The Symmetry-Adapted Linear Combinations for the Molecular Orbitals of H2+ and H2 -- 7.2.2 The Chemical Bond Energy from Molecular Orbitals -- 7.2.3 The Molecular Orbital Energy -- 7.2.4 Bond Order -- 7.3 The Relative Energies of Hydrogen-Like Atomic Orbitals -- 7.3.1 Radial Behaviour of Atomic Orbitals -- 7.3.2 The Relative Energies of Atomic Orbitals in Different Elements -- 7.3.3 The Relative Energies of Atomic Orbitals from Electronegativity -- 7.4 The Molecules Formed by Other Second-Row Elements with Hydrogen -- 7.4.1 BeH2, Beryllium Hydride -- 7.4.2 BH3, Boron Hydride -- 7.4.3 CH4, Methane -- 7.4.4 NH3, Ammonia -- 7.4.5 H2O, Water -- 7.5 The Second-Row Diatomic Molecules -- 7.5.1 Homonuclear Diatomics -- 7.5.2 Heteronuclear Diatomics of Second-Row Elements -- 7.6 More Complex Polyatomic Molecules -- 7.6.1 Ethene -- 7.7 Metal Complexes.

7.7.1 Complexes Containing σ-Donor Ligands -- 7.7.2 The Jahn-Teller Effect -- 7.7.3 Complexes Containing Ligand Orbitals of π-Symmetry -- 7.8 Summary -- 7.9 Self-Test Questions -- Further Reading -- Appendix 1 H2O Models for Identifying the Results of Symmetry Operation Products -- Appendix 2 Assignment of Chiral Centre Handedness using Cahn-Ingold-Prelog Rules -- Appendix 3 Model of a Tetrahedron and the Related Cube -- Appendix 4 Model of an Octahedron -- Appendix 5 Matrices and Determinants -- A5.1 Matrices as Representations of Symmetry Operators -- A5.1.1 Products of Matrices -- A5.1.2 Products of Matrices, Expressed as Summations -- A5.2 Matrices for Solving Sets of Linear Equations -- Further Reading -- Appendix 6 The Mathematical Background to Infrared Selection Rules -- A6.1 Model Based on Classical Mechanics -- A6.2 Model Based on Quantum Mechanics -- A6.3 Excited Vibrational States -- A6.4 Vibrational Modes for Polyatomic Molecules -- A6.5 Generalization to Arbitrary Transitions -- A6.6 Summary of Selection Rules -- Further Reading -- Appendix 7 The Franck-Condon Principle -- Appendix 8 Classical Treatment of Stokes/Anti-Stokes Absorption -- Appendix 9 The Atomic Orbitals of Hydrogen -- A9.1 Choice of Coordinate System -- A9.2 Separation of Variables -- A9.3 The Angular Equation -- A9.4 Physical Interpretation of the Angular Equation Solutions -- A9.5 Angular Momentum -- A9.6 The Radial Equation -- A9.7 The Complete Atomic Orbitals -- A9.8 Expectation Values -- A9.9 Real Combinations to Form the Familiar Atomic Orbitals -- A9.10 Cartesian Forms of the Real Angular Functions -- A9.11 Endnote on Imaginary Numbers -- Further Reading -- Appendix 10 The Origin of Chemical Bonding in H2+ -- A10.1 Chemical Bond Formation -- A10.2 H Atom and H+ Cation -- A10.3 The Virial Theorem -- A10.4 H2+ Molecule.

A10.5 Choice of Coordinate System for H2+: Cylindrical Polar Coordinates -- A10.6 H2+: the Electron Kinetic Energy -- A10.7 H2+: the Electronic Potential Energy -- A10.8 The Chemical Bond Formation Energy Based on Rigid Atomic Orbitals -- A10.9 Optimal Radial Decay of Molecular Orbitals -- Further Reading -- Appendix 11 H2O Molecular Orbital Calculation in C2v Symmetry -- Further Reading -- Appendix 12 Character Tables -- A12.1 Non-Axial Groups -- A12.2 Axial Groups -- A12.2.1 Cn Groups -- A12.2.2 Sn Groups -- A12.2.3 Cnv Groups -- A12.2.4 Cnh Groups -- A12.2.5 Dn Groups -- A12.2.6 Dnd Groups -- A12.2.7 Dnh Groups -- A12.3 Cubic Groups -- A12.3.1 Tetrahedral, Td -- A12.3.2 Rotational Subgroup of Td, T -- A12.3.3 Octahedral, Oh -- A12.3.4 Rotational Subgroup of Oh,O -- A12.4 Groups for Linear Molecules -- Index.